QUASICONFORMAL MAPPINGS IN AHLFORS REGULAR SPACES

IN AHLFORS REGULAR SPACES

ANCA ANDREI

The problem on continuous or homeomorphic extension to boundary of the quasi- conformal mappings in Euclideann-space of dimension at least two is well-known.

It is natural to ask whether the boundary extension theorems are valid in more abstract setting, i.e., the AhlforsQ-regular metric measure spaces (Q≥1). Those the three (metric, geometric and analytic) definitions of quasiconformality in Eu- clidean spaces can be formulated in this setting but they are not equivalent. Our interest in this talk is in the notion of geometric quasiconformality. We deal with the extension to boundary for geometrically quasiconformal mappings on domains in AlhforsQ-regular spaces. For this, we define a number of concepts allowing us to describe the properties of a domain at a boundary point. We give several boundary extension theorems for geometrically quasiconformal mappings.

AMS 2000 Subject Classification: 30C65.

Key words: quasiconformal mapping, AhlforsQ-regular space, locally connected, finitely connected, m-connected, very weakly flat, property P1, pro- pertyP2^∗, strictly quasiconformal accessibility.

We first recall some definitions and make some remarks of a general character which will be used in this paper.

LetX be a topological space. The spaceX is said to be path-connected (or pathwise connected or 0-connected) if for any two pointsxandyinXthere exists a continuous function f from the unit interval [0,1] to X withf(0) =x and f(1) = y. (This function is called a path from x toy.) The space X is said to be arc-connected (or arcwise connected) if any two distinct points can be joined by an arc, that is a path γ which is a homeomorphism between the unit interval [0,1] and its imageγ([0,1]). It can be shown that any Hausdorff space which is path-connected is also arc-connected. The spaceXis said to be rectifiable connected if any two points can be joined by a rectifiable path. A domain DinX is an open connected non-empty set inX. An arc domainD inX is an open arc-connected set inX. By a continuum we mean a compact connected set which contains at least two points. A neighbourhood of a setE is an open set containing E.

REV. ROUMAINE MATH. PURES APPL.,54(2009),5–6, 361–373

The purpose of this paper is to give some results concerning the local and global boundary behaviour of quasiconformal mappings on domains in Ahlfors regular metric measure spaces. These will be done in terms of the properties possessed by the domains in question.

We therefore recall and introduce the concepts associated with the boun- dary of a domain in a metric space. Also, we give some relationships between these concepts. The standard references for Euclidean case are in [1], [2], [5], [7] and for metric case in [4]. Throughout the paper we shall consider only metric spaces. Further, all metric spaces are assumed to be rectifiable con- nected and all measures are assumed to be locally finite and Borel regular with dense support.

We shall denote by X or Y a metric measure space that satisfies the standard assumptions stated above.

Let us consider a metric measure space (X, d, µ), a domain DinX and a boundary point bof D.

Definition 1. D is locally (arc) connected at b ifb has arbitrarily small neighbourhoods U such that U ∩D is (arc) connected. This means that for each neighbourhood V of bthere exists a neighbourhood U of b U ⊂V such that U ∩D is (arc) connected.

Definition 2. D is finitely (arc) connected at bif bhas arbitrarily small neighbourhoods U such that U∩Dhas a finite number of (arc) components.

Definition3. Dism-(arc) connected atb,m= 1,2, . . ., ifbhas arbitrarily small neighbourhoods U such thatU ∩Dconsists of m (arc) components.

Obviously, a domain is locally (arc) connected at a boundary point if and only if it is 1-(arc) connected at the point.

Note that m-(arc) connectedness always implies finite (arc) connected- ness. The converse is not true.

In the sequel, we consider a family Γ of (nonconstant) paths in X. A Borel function ρ : X → [0,∞] is admissible for Γ if R

γρds ≥ 1 for all lo- cally rectifiable paths γ in Γ. The set of all admissible functions for Γ is denoted by F(Γ).

Definition 4. The (conformal)p-modulus of Γ, 1≤p <∞, is defined as:

Mp(Γ) = inf

ρ∈F(Γ)

Z

X

ρ^pdµ.

Note, in particular, that the modulus of the collection of all non-locally rectifiable paths is zero.

Definition 5. A metric measure space (X, d, µ) is said to be (Ahlfors) Q-regular, Q >0, if there exists a constant c <∞ such that

r^Q

c ≤µ(B_r)≤cr^Q for every ball B_r inX with radiusr <diamX.

Note that if the above relation holds, thenXhas Hausdorff dimensionQ.

IfE, F, Dare subsets ofXwithE⊂D,F ⊂D, we denote by ∆(E, F;D) the family of all paths which join E and F inD.

Next, we only considerQ-regular metric measure spaces with 1≤Q <∞ and a domainDinX. If Γ is a family of paths inX, we writeM(Γ) =M_Q(Γ).

We now introduce the

Definition 6. D is (arc) very weakly flat at a point b∈∂D if for every number δ > 0 there exist arbitrarily small neighbourhoods U of b such that M(∆(E, F;D))≥δ for all (arc) connected sets E, F ⊂ D withE∩∂U 6=∅, F ∩∂U 6=∅.

More precisely, by the phrase “there exist arbitrarily small neighbour- hoodsU ofb” in the above definition we understand that for every neighbour- hood U of b there is a neighbourhood V of b, V ⊂U, such that M(∆(E, F; D))≥δ for all (arc) connected sets E, F ⊂D withE∩∂V 6=∅,F ∩∂V 6= 0 and E∩∂U 6=∅,F∩∂U 6=∅.

This definition is different from the definition of a domain that is weakly flat at a boundary point in [4, 13.3]. Moreover, if a domain is very weakly flat at a boundary point, then it is weakly flat at the point.

The property P₁ (see [7], Definition 17.5.(3)), the property P₂^∗ (see [1], Definition 5) and quasiconformal accesibility (see [5], 1.7(vii)) in the Euclidean space can be formulated in the same way in our framework.

Definition 7. We say thatDhas (arc) property P₁ (or is (arc) quasicon- formally flat) at b ∈ ∂D if M(∆(E, F;D)) = ∞ for all (arc) connected sets E, F ⊂D withb∈E∩F.

Definition 8. We say that D has (arc) property P₂^∗ at b ∈ ∂D if for every point b1 ∈∂D1, b1 6=b, there exist a continuumF ⊂D and a number δ > 0 such that M(∆(E, F;D))≥δ for all (arc) connected sets E ⊂Dwith b, b₁∈E.

Definition 9. We say that D is (arc) quasiconformally accessible atb∈

∂D if for every neighbourhood U of b there exist a continuum F ⊂ D and a positive number δ >0 such that M(∆(E, F;D))≥δ for all (arc) connected setsE ⊂D withb∈E andE∩∂U 6=∅.

Now, we introduce

Definition 10. We say thatDis (arc) strictly quasiconformally accessible atb∈∂D if there exist arbitrarily small neighbourhoods U ofb, a continuum

F in D, and a real number δ > 0 such that M(∆(E, F;D)) ≥ δ for every E ⊂D (arc) connected, with E∩∂U 6=∅, i.e., for every neighbourhoodU of b there exist a neighbourhoodV of b V ⊂U, a continuumF inDand a real numberδ >0 such thatM(∆(E, F;D))≥δ for everyE⊂D(arc) connected with E∩∂V 6=∅and E∩∂U 6=∅.

We mention that the definition of strictly quasiconformal accessibility is different from the definition of strict accessibility [4, 13.3]. Moreover, if a domain is strictly quasiconformal accessible at a boundary point, then it is strictly accessible at the point.

Note that (arc) strictly quasiconformal accessibility implies (arc) qua- siconformal accessibility. This claim follows by Definitions 9, 10 and result below.

Lemma1. Let Ebe a connected subset of a topological space X.If A⊂X and neither E∩A nor E∩(X\A) is empty, then E∩∂A6=∅[8, 11.26].

Proof. IfE∩∂A=∅ then∅=E∩(A∩(X\A)) =E∩A∩(X\A) and E =E∩X=E∩(A∪(X\A)) =E∩(A∪(X\A)) = (E∩A)∪(E∩(X\A)) which contradict the connectedness of E.

Next, we give two propositions which establish some relationships bet- ween the notions introduced above.

Proposition 1. Suppose that D is an arc domain. If D is (arc) very weakly flat at b∈∂D, then D is(arc) strictly quasiconformal accessible at b.

Proof. LetU be a neighbourhood ofband a numberδ >0. We consider a ball B(b, r) ⊂ U, 0 < r < sup

x∈D

d(b, x) and 0 < r₀ < r. By the (arc) very weakly flatness of D, there exists a neighbourhood V of b,V ⊂B(b, r0) such that M(∆(E, F;D))≥δ for every (arc) connected sets E, F ⊂D, with E∩∂V 6=∅,F∩∂V 6=∅,E∩∂B(b, r₀)6=∅,F∩∂B(b, r₀)6=∅. SinceD is an arc domain andV an open set, there existx1 ∈D∩∂B(b, r₀) andx2 ∈∂V ∩D.

We choose as a continuum F an arbitrary curve connecting the pointsx₁ and x2 in D. Hence M(∆(E, F;D)) ≥ δ for every (arc) connected set E ⊂ D, with E ∩∂V 6= ∅ and E ∩∂B(b, r₀) 6= ∅. On the other hand, every (arc) connected set E inDthat intersects ∂U and ∂V will also intersect∂B(b, r0).

Consequently, M(∆(E, F;D))≥δ for every (arc) connected set E ⊂Dwith E∩∂V 6=∅,E∩∂U 6=∅. Thus, we get the desired conclusion.

Proposition2. Suppose that D is an arc domain. If Dis very weakly flat at b∈∂D,then Dis locally arc connected at b.

Proof. SinceD is very weakly flat atb, it is weakly flat atb. By Lemma 13.18 in [4], Dis locally arc connected at band the proof is complete.

Since locally arc connectedness implies local connectedness, we deduce that if Dis very weakly flat at b∈∂D, thenDis locally connected at b.

The following abbreviation will be used in this paper: if a domain has one of the properties in Definitions 1 through 3, 6 through 10 at each boundary point, then it is said to have the property in question on the boundary.

We consider twoQ-regular metric measure spaces (X, d, µ) and (Y, d⁰, µ⁰) with 1≤Q <∞, and two domainsD⊂X,D⁰ ⊂Y. Similarly to the geometric definition in Rⁿ,n≥2, by V¨ais¨al¨a [7, 13.1] we have:

Definition11. A homeomorphismf :D→D⁰is calledK-(geometrically) quasiconformal, K∈[1,∞], if

M(Γ)

K ≤M(f(Γ))≤KM(Γ)

for every family Γ of paths inD. We also say that a homeomorphismf :D→ D⁰ is (geometrically) quasiconformal iff isK-(geometrically) quasiconformal for someK ∈[1,∞), i.e., if the distortion of moduli of path families under the mapping f is bounded.

To simplify notation, we write quasiconformal mapping instead of geome- trically quasiconformal mapping.

In the sequel the notation f : D → D⁰ includes the assumption that D⊂X,D⁰ ⊂Y and D, D⁰ are arc domains.

Theorem1. Suppose that f :D→D⁰ is a quasiconformal mapping,D⁰ is compact and D is(arc) very weakly flat at a point b∈∂D. If D⁰ is finitely (arc) connected on the boundary, then there exists the limit of f at b.

Proof. First, we prove that the cluster set C(f, b) off at b contains at most one point at which D⁰ is finitely connected. Note that C(f, b) 6= ∅ by the compactness of D⁰, and that C(f, b) belongs to the boundary of the set D⁰ sincef is a homeomorphism between two open sets. Suppose that D⁰ is finitely (arc) connected at two distinct pointsb⁰₁, b⁰₂ ∈C(f, b). We choose balls B1 =B(b⁰₁, r1), B2 =B(b⁰₂, r2) such that B1∩B2=∅.

By the definition of a cluster set, there exist two sequences (x_j),(y_j), such that xj, yj ∈D, xj →b, yj →b and f(xj)→ b⁰₁, f(yj) → b⁰₂. Since D⁰ is finitely arc connected at b⁰₁, there exists a neighbourhoodV₁ ofb⁰₁, V₁ ⊂B₁, such that V1∩D⁰ has a finite number of (arc) components. Therefore, there is a subsequence of (f(x_j)) which is contained in a single (arc) component of V₁ ∩D⁰, hence in a single (arc) component of B₁ ∩D⁰. Let E₁ be an (arc) component of B1 ∩D⁰ which contains a subsequence of (f(xj)). Similary, B₂∩D⁰ has an (arc) component E₂ which contains a subsequence of (f(y_j)).

The subsets f⁻¹(E₁), f⁻¹(E₂) of D are (arc) connected and b ∈ f⁻¹(E₁)∩ f⁻¹(E2). Set Γ = ∆(f⁻¹(E1), f⁻¹(E2);D) and Γ⁰ = ∆(E1, E2;D⁰). Let us

consider r0 ∈(0, d0),d0 = sup

x∈D

d(b, x) and B =B(b, r), 0< r < r0, such that f⁻¹(E₁)∩∂B6=∅,f⁻¹(E₂)∩∂B6=∅. We can choosersuch: takea₁ ∈f⁻¹(E₁), a2 ∈f⁻¹(E2). Setd1 =d(b, a1),d2 =d(b, a2). We take 0< r <min{r₀, d1, d2}.

By the choise of r we have a₁, a₂ ∈/ B(b, r), b ∈ B(b, r) and it follows from Lemma 1 that f⁻¹(E1)∩∂B6=∅,f⁻¹(E2)∩∂B 6=∅.

By the (arc) very weakly flatness of D at b, for every δ > 0 and U = B(b, r) there exists V = B(b, r∗) with 0 < r∗ < r such thatM(Γ) ≥δ since f⁻¹(E1)∩∂D(b, r∗) 6= ∅, f⁻¹(E2)∩∂B(b, r∗) 6= ∅ and f⁻¹(E1)∩∂B 6= ∅, f⁻¹(E₂)∩∂B 6=∅.

The quasiconformality of f implies that there exists a constant K <∞ such that

(1) M(Γ⁰)≥ M(Γ)

K ≥ δ K

for every δ >0. Takeh=d⁰(b⁰₁, b⁰₂).Since D⁰ is compact, we haved <∞and, by Lemma 2.7 in [4],

(2) M(Γ⁰)≤ µ⁰(D⁰)

h^Q₁ , where h1 =h−r1−r2.

Sinceµ⁰ is locally finite andD⁰is compact, there exists a constantC >0 such that µ⁰(D⁰)≤C and by (2) we get

(3) M(Γ⁰)≤ C

h^Q₁ .

Choosing δ > ^KC

h^Q₁ , by (1) and (3) we reach a contradiction. Consequently, C(f, b) contains at most one point b⁰ at which D⁰ is finitely (arc) connected.

On the other hand, D⁰ is finitely (arc) connected on the boundary, hence

x→blimf(x) =b⁰. Thus, the proof is complete.

Using an argument similar to that employed in the proof of Theorem 1 one can show that Theorem 17.15 in [7] in the Euclidean n-space also holds in our framework.

Theorem2. Suppose that f :D→D⁰ is a quasiconformal mapping,D⁰ is compact and D has (arc) property P1 at a point b ∈∂D. If D⁰ is finitely (arc) connected on the boundary, then there exists the limit of f at b.

Proof. Consider sets E₁, E₂, f⁻¹(E₁), f⁻¹(E₂),Γ and Γ⁰ as in the proof of Theorem 1. Since b ∈ f⁻¹(E1)∩f⁻¹(E2) and the sets f⁻¹(E1), f⁻¹(E2) are (arc) connected, by the (arc) property P₁ of D at b we haveM(Γ) =∞.

Quasiconformality of f implies thatM(Γ⁰) =∞. Takingh1 as in the proof of

Theorem 1, we haveM(Γ⁰)≤ ^µ0(D0)

h^Q₁ <∞, that contradicts the above equation.

Consequently, we obtain the desired conclusion.

Using Theorems 1 and 2 we obtain the important results below.

Corollary1. If D is (arc) very weakly flat (or has (arc) property P₁) on the boundary, D⁰ is finitely (arc) connected on the boundary and D⁰ is compact, then every quasiconformal mapping f : D → D⁰ has a continuous extension f^∗ :D→D⁰.

Corollary 2. If D is (arc) very weakly flat (or has (arc) property P₁) on the boundary, D⁰ is locally(arc) connected on the boundary and D⁰ is compact, then every quasiconformal mapping f : D → D⁰ has a continuous extension f^∗ :D→D⁰.

Corollary 3. Suppose that D and D⁰ are (arc) very weakly flat on the boundary and D, D⁰ are compact. Then every quasiconformal mapping f :D→D⁰ has a homeomorphic extension f^∗:D→D⁰.

Proof. The result is an immediate corollary of Proposition 2 and Corol- lary 2.

In what follows we suppose thatQ >1.

The following theorem is an analogue of Theorem 17.15 in [7] in the Euclidean case. Remark that instead of property P2 one considers (arc) prop- erty P₂^∗.

Theorem3. Suppose that f :D→D⁰ is a quasiconformal mapping and D⁰ is compact. If D is locally (arc) connected at b ∈ ∂D and D⁰ has (arc) property P₂^∗ at least one point of C(f, b),then there exists the limit of f at b.

Proof. Assume that C(f, b) contains two distinct points b⁰₁, b⁰₂ and that D⁰ has (arc) property P₂^∗ atb⁰₁. By the definition of (arc) property P₂^∗, there exist a continuum F ⊂D⁰ and a positive number δ >0 such that

(4) M(∆(E, F;D⁰))≥δ

whenever E is (arc) connected inD⁰ withb⁰₁, b⁰₂ ∈E. We considerr₀ ∈(0, d₀) with d₀ = sup

x∈D

d(x, b) and a natural number n₀ such that 0 < _n¹

0 < r₀. Since D is locally (arc) connected at b, there is a neighbourhoodVn0 of bsuch that V_n0 ⊂B_n0 =B(b,_n¹

0) andV_n0∩Dis (arc) connected. But there exists a ball B_n1 =B(b,_n¹

1) ⊂V_n0 withn₁ > n₀, where n₁ is a natural integer. Applying the same procedure we choose a sequence V_n0, V_n1, . . . of neighbourhoods ofb such that every Cni = Vni ∩D is (arc) connected, open and Bni+1 ⊂ Vni ⊂ Bni for all i ≥ 0. Set Γi = ∆(Cni, f⁻¹(F);D) and Γ⁰_i = ∆(f(Cni), F;D⁰).

Moreover, dist(b, Cni)→0 asi→ ∞. Sincef is a homeomorphism,f(Cni) is (arc) connected, open andf⁻¹(F) is a continuum. Moreover, by the inclusion C(f, b)⊂f(V_ni ∩D) =f(C_ni) we haveb⁰₁, b⁰₂∈f(C_ni). Using (4) we get

(5) M(Γ⁰_i)≥δ

for any i. By the quasiconformality off, there exists a constant K >0 such that M(Γ⁰_i)≤K·M(Γi) and therefore

(6) M(Γ_i)≥ δ

K

for any i. Sincef⁻¹(F) is a continuum, f⁻¹(F)⊂D,Dis open, andb∈∂D, we haveb /∈f⁻¹(F) andB_ni∩f⁻¹(F) =∅forilarge, henceC_ni∩f⁻¹(F) =∅.

Fix such an i and denote r_i = _n¹

i. Every path connecting f⁻¹(F) and C_ni+j intersects ∂Bni and ∂Bni+j for j ≥ 1. But rk → 0 as k → ∞, hence 0 <

2r_i+j < r_i <∞ forj large.

By Lemma 3.14 in [3], there exists a constant C₀ only depending on Q and a constant C (by the Q-regularity of X) such that

(7) M(Γi+j)≤C0

log ri

r_j+i 1−Q

.

Since (log_r^ri

j+i)^1−Q → 0 as j → ∞, inequality (7) contradicts (6). Conse- quently, C(f, b) has at most one point for which D⁰ has (arc) property P₂^∗, hence we get the desired conclusion.

In the same manner we can prove the next result.

Theorem4. Suppose that f :D→ D⁰is a quasiconfomal mapping and D⁰ is compact. If Dis locally(arc)connected at b∈∂Dand D⁰ is(arc)strictly quasiconformally accessible at least one point of C(f, b),then there exists the limit of f at b.

Proof. Suppose thatC(f, b) contains two distinct points b⁰₁, b⁰₂ and that D⁰ is (arc) strictly quasiconformally accessible at b⁰₁. Let U be a neighbour- hood of b⁰₁ such that b⁰₂ ∈/ U. By the definition of (arc) strict quasiconformal accesibility of D⁰ atb⁰₁ there exist a neighbourhood V of b⁰₁, V ⊂U, a conti- nuum F ⊂D⁰ and a numberδ >0 such that

(8) M(∆(E, F;D⁰))≥δ

whenever E ⊂ D⁰ is (arc) connected, with E ∩∂V 6= ∅, E ∩∂U 6= ∅. As in the proof of Theorem 3, we consider the sets Γi and Γ⁰_i. Since b⁰₁, b⁰₂ ∈ f(Cni), f(Cni) is (arc) connected, andb⁰₂ ∈/ U, by Lemma 1 we havef(Cni)∩

∂V 6=∅,and f(C_ni)∩∂U 6=∅. Inequality (8) implies

(9) M(Γ⁰_i)≥δ

for any i. Finally, we use the arguments from the proof of Theorem 3 and obtain the desired conclusion.

Corollary 4. Suppose that f : D → D⁰ is a quasiconfomal mapping and D⁰ is compact. If D is locally (arc) connected at b∈∂D and D⁰ is (arc) very weakly flat at least one point of C(f, b), then there exists the limit of f at b.

Proof. It follows from Theorem 4 and Proposition 1.

Corollary 5. If D is locally (arc) connected on the boundary, D⁰ is compact and D⁰ is(arc) strictly quasiconformally accessible[or has(arc) pro- perty P₂^∗] on the boundary, then every quasiconformal mapping f :D→ D⁰ has a continuous extension f^∗:D→D⁰.

Proof. It follows from Theorems 3 and 4.

Corollary 6. If D is (arc) very weakly flat on the boundary, D⁰ is compact and D⁰ is(arc)strictly quasiconformally accessible[or has(arc) pro- perty P₂^∗]on the boundary, then every quasiconformal mapping f :D→ D⁰ has a continuous extension f^∗ :D→D⁰.

Proof. It is an immediate corollary of Proposition 2 and Corollary 5.

Corollary 7. Suppose that f :D → D⁰ is a quasiconformal mapping and D, D⁰ are compact sets. If D, D⁰ are locally (arc) connected on the boun- dary and (arc) strictly quasiconformally accessible [or has (arc) property P₂^∗] on the boundary, then f can be extended to a homeomorphism f^∗ :D→D⁰.

Proof. It follows from Theorem 4 and Corollary 4.

In order to prove the next theorem we use the result below that for the Euclidean case is proved in Theorem 1.10 in [5].

Proposition 3. Suppose that D is m-(arc) connected at b ∈ ∂D and (b_1,k), . . . ,(b_m+1,k) are m+ 1 sequences of points in D with limit b. If U is a neighbourhood of b,then there exists a component of U∩Dwhich contains subsequences of two different sequences.

Proof. Let (b1,k), . . . ,(bm+1,k) be m+ 1 sequences of points of D with limit b, and letU be a neighbourhood of b.

By m-connectedness of D at b, there exists a neighbourhood V of b, V ⊂ U, and V ∩D =E1∪. . .∪Em, where Ei is a component of V ∩D for any i∈ {1, . . . , m}. Sinceb_j,k →bask→ ∞, for anyj∈ {1, . . . , m+ 1}there exists a natural numberk0such thatbj,k ∈V∩Dwheneverk≥k0. Therefore, there exists E_j1 which contains a subsequence (b⁰_1,k) of (b_1,k). Using the same reasoning for the sequences (b_2,k), . . . ,(b_m+1,k) we findE_j2, . . . , E_jm+1, each of

them containing one subsequence (b⁰_2,k), . . . ,(b⁰_m+1,k). Since {j₁, . . . , jm+1}= {1, . . . , m}, at least oneEi, i∈ {1, . . . , m}, contains subsequences of two dif- ferent sequences mentioned above. These subsequences are contained in a single component of U ∩D. The proof is complete.

We shall prove that Theorem 2 in [2] established in the Euclidean case holds in our framework.

Theorem 5. Suppose that f : D → D⁰ is a quasiconformal mapping and that D, D⁰ are compact sets. If D is m-(arc) connected at b∈∂D, then C(f, b) either contains at most m−1 points at which D⁰ has (arc) property P₂^∗ or consists of m points.

Proof. Suppose, contrary to the assertion, that C(f, b) contains at least m + 1 distinct points b⁰₁, . . . , b⁰_m+1 and that D⁰ has (arc) property P₂^∗ at b⁰₁, . . . , b⁰_m. Since D⁰ has (arc) property P₂^∗ at b⁰_i for each i ∈ {1, . . . , m}, forb⁰_j,j6=i,j∈ {1, . . . , m+ 1}, there exist a continuumA⁰_ij inD⁰ and a real number δij >0 such that

(10) M(∆(F⁰, A⁰_ij;D⁰))≥δij

whenever F⁰ is an (arc) connected set inD⁰ withb⁰_i, b⁰_j ∈F⁰ .

Let δ = min{δ_ij, 1 ≤ i ≤ m, 1 ≤ j ≤ m+ 1, i 6= j}, Aij = f⁻¹(A⁰_ij), d_ij =d(A_ij, ∂D), d= min{d_ij,i6=j, 1≤i≤m, 1≤j≤m+ 1}.

Fixε, 0<2ε < d. Since b⁰_j ∈C(f, b) for eachj∈ {1, . . . , m+ 1}, we can choose a sequence (bjk) inD, bjk →b ask→ ∞and f(bjk)→b⁰_j. Since Dis m-(arc) connected at b, by Proposition 3, there exist an (arc) component F of B(b, ε)∩D and 1≤i < j ≤m+ 1 such that F contains two subsequences of (b_ik)_k and (b_jk)_k. The set F⁰ = f(F) is (arc) connected and b⁰_i, b⁰_j ∈ F⁰. Taking Γij = ∆(Aij, F;D), Γ⁰_ij = ∆(A⁰_ij, F⁰;D⁰) and using (10) we get

(11) M(Γ⁰_ij)≥δ.

On the other hand, every path γ ∈Γ_ij intersects both ∂B(b, d) and ∂B(b, ε) and by Lemma 3.14 in [3] there exists a constantC0 only depending onQand a constant C (by Q-regularity of X) such that

(12) M(Γ_ij)≤C₀

logd

ε 1−Q

.

Since (12) holds for all 0 < 2ε < d, letting ε → 0, using (11) and the fact that f is a quasiconformal mapping, we reach a contradiction. The proof is complete.

The next result offers an analogue of Theorem 5 in the case of (arc) strictly quasiconformal accessibility.

Theorem 6. Suppose that f : D → D⁰ is a quasiconformal mapping and that D, D⁰ are compact sets. If D is m-(arc) connected at b ∈∂D, then C(f, b) either contains at most m −1 points at which D⁰ is (arc) strictly quasiconformally accessible or consists of m points.

Proof.Suppose thatC(f, b) contains at leastm+ 1 distinct pointsb⁰₁, . . . , b⁰_m+1 and that D⁰ is (arc) strictly quasiconformally accessible at b⁰₁, . . . , b⁰_m. Since D⁰ is (arc) strictly quasiconformally accessible atb⁰_i for each i∈ {1, . . . , m}, for any neighbourhood Ui of b⁰_i there exist a neighbourhood Vi of b⁰_i, V_i ⊂U_i, a continuum A⁰_i in D⁰ and a real number δ_i >0 such that

(13) M(∆(F⁰, A⁰_i;D⁰))≥δi

whenever F⁰ is an (arc) connected set in D⁰ withF⁰∩∂U_i 6=∅,F⁰∩∂V_i 6=∅.

We can choose U_i such that b⁰_j ∈/ U_j for all j 6= i, j ∈ {1, . . . , m+ 1}. Set δ = min

1≤i≤mδ_i, A_i =f⁻¹(A⁰_i), d_i =d(A_i, ∂D), d= min

1≤i≤md_i. Let εbe such that 0 < 2ε < d. Since b⁰_j ∈ C(f, b) for each j ∈ {1, . . . , m+ 1}, we can choose a sequence (b_jk) in D, b_jk → b ask → ∞ and f(b_jk) → b⁰_j . By Proposition 3 there exist a component F ofB(b, ε)∩D and 1≤i < j ≤m+ 1 such thatF contains two subsequences of (b_ik) and (b_jk).

The set F⁰ =f(F) is connected and b⁰_i, b⁰_j ∈ F⁰. Since b⁰_i ∈ V_i, b⁰_j ∈/ V_i and b⁰_i ∈U_i,b⁰_j ∈/ U_j forj6=i, we haveF⁰∩∂V_i6=∅andF⁰∩∂U_j 6=∅. Taking Γ_i= ∆(A_i, F;D), Γ⁰_i= ∆(A⁰_i, F⁰;D⁰) and using (13) we obtain

(14) M(Γ⁰_i)≥δ.

But, every path γ∈Γ_i meets both∂B(b, d) and ∂B(b, ε) and by Lemma 3.14 in [3] there exists a constant C0 such that

(15) M(Γ_i)≤C₀

logd

ε 1−Q

.

Lettingε→0 and using (14) and the fact thatf is a quasiconformal mapping, we reach a contradiction, which completes the proof.

Finally, we show that Theorem 3 in [2] holds in our setting, too.

Theorem7. Let f :D→D⁰ be a quasiconformal mapping and suppose that D is m-(arc) connected at b ∈∂D. If D⁰ is compact and D⁰ has (arc) property P₂^∗ [or D⁰ is (arc) strictly quasiconformally accessible] at each point of C(f, b)and C(f, b) is a connnected set, then there exists the limit of f at b.

Proof. If m= 1 thenD is 1-(arc) connected atb and, by Theorem 5 [or Theorem 6], C(f, b) either contains no point at which D⁰ has (arc) property P₂^∗ [or it is (arc) strictly quasiconformally accessible], or C(f, b) has a single

point. Since D⁰ has (arc) propertyP₂^∗ [or it is (arc) strictly quasiconformally accessible] at each point of C(f, b), the latter has a single point.

Assume that m ≥2. By Theorem 5 [or Theorem 6], C(f, b) either con- tains at most m−1 points at which D⁰ has (arc) property P₂^∗ [or is (arc) strictly quasiconformally accessible] or consists of m points. Since C(f, b) is connected, C(f, b) has at most m−1 points at which D⁰ has (arc) property P₂^∗ [or is (arc) strictly quasiconformally accessible]. IfC(f, b) does not have a single point, since C(f, b) is a connected set, there exists an infinity of points in C(f, b). By hypothesis, D⁰ has (arc) property P₂^∗ [or it is (arc) strictly quasiconformally accessible] at these points. This contradicts the fact that C(f, b) has at most m−1 points at which D⁰ has (arc) propertyP₂^∗ [or it is (arc) strictly quasiconformally accessible].

Corollary 8. Let f :D→ D⁰ be a quasiconformal mapping and sup- pose that Dis m-(arc) connected at b∈∂D. If D⁰ is compact and D⁰ is(arc) very weakly flat at each point of C(f, b) and C(f, b) is a connected set, then there exists the limit of f at b.

Proof. It follows from Proposition 1 and Theorem 7.

According to the discussion above, if one considers the metric spaces which have a base of topology consisting of path-connected sets (arc) locally path-connected, then instead of arc domains we can consider domains.

Note that if X and Y are proper Q-regular, Q-Loewner spaces, Q >1, then we can consider quasiconformal mappings in any sense (metric, geome- tric or analytic) since the three definitions of quasiconformality are equiva- lent (see [6]).

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Received 20 January 2009 “Spiru Haret” National College of Computer Science

17 Zorilor St.

Suceava, Romania anca56@hotmail.com